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Friday, November 30, 2012 at 3:30pm
Take a digital photo of a natural scene. For simplicity, convert the photo from color to black and white. The photo can be reduced, or scaled, to make a new (smaller) picture, say half the size in both dimensions. The new picture is of a scene in which each of the original objects, and in fact every imaged point, has been relocated twice as far from the camera. This “stretching” is artificial in that it does not correspond to any movement of the camera in the real world. Yet the picture looks perfectly normal, and the local spatial statistical structure (e.g. the distribution of values of horizontal or vertical derivatives) is indistinguishable from the local spatial statistical structure of the original. “Images of natural scenes are scale invariant.”
Pick a stock or a collection of stocks and make a histogram of the returns using, say, ten years of five-minute intervals. Now do the same for the same stock or group of stocks, but using anything from one-minute returns to one-thousand-minute returns. A single scaling parameter, independent of the return interval, renders all of these histograms indistinguishable. “Stock returns are self-similar.”
Mathematical models of images, or more generally of spatial processes, are never scale invariant unless they are trivial (e.g. constant gray levels) or exotic (lacking a direct definition in terms of image intensities, i.e. “random measures”). And state-of-the-art models of market fluctuations are not self-similar. The sources of these invariants are an enduring mystery. I will propose some explanations and make connections to perception and coding in images, and to volatility and regularities of large returns in the markets.